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Оглавление
- Preface
- Contents
- 1 Probabilities
- 1.1 Probability Spaces
- 1.1.1 Sample Spaces
- 1.1.2 3-Fields of Events
- 1.1.3 Probability Measures
- 1.2 Basic Properties of Probability Measures
- 1.3 Discrete Probability Measures
- 1.4 Special Discrete Probability Measures
- 1.4.1 Dirac Measure
- 1.4.2 Uniform Distribution on a Finite Set
- 1.4.3 Binomial Distribution
- 1.4.4 Multinomial Distribution
- 1.4.5 Poisson Distribution
- 1.4.6 Hypergeometric Distribution
- 1.4.7 Geometric Distribution
- 1.4.8 Negative Binomial Distribution
- 1.5 Continuous Probability Measures
- 1.6 Special Continuous Distributions
- 1.6.1 Uniform Distribution on an Interval
- 1.6.2 Normal Distribution
- 1.6.3 Gamma Distribution
- 1.6.4 Exponential Distribution
- 1.6.5 Erlang Distribution
- 1.6.6 Chi-Squared Distribution
- 1.6.7 Beta Distribution
- 1.6.8 Cauchy Distribution
- 1.7 Distribution Function
- 1.8 Multivariate Continuous Distributions
- 1.8.1 Multivariate Density Functions
- 1.8.2 Multivariate Uniform Distribution
- 1.9 *Products of Probability Spaces
- 1.9.1 Product 3-Fields and Measures
- 1.9.2 Product Measures: Discrete Case
- 1.9.3 Product Measures: Continuous Case
- 1.10 Problems
- 1.1 Probability Spaces
- 2 Conditional Probabilities and Independence
- 2.1 Conditional Probabilities
- 2.2 Independence of Events
- 2.3 Problems
- 3 Random Variables and Their Distribution
- 3.1 Transformation of Random Values
- 3.2 Probability Distribution of a Random Variable
- 3.3 Special Distributed Random Variables
- 3.4 Random Vectors
- 3.5 Joint and Marginal Distributions
- 3.5.1 Marginal Distributions: Discrete Case
- 3.5.2 Marginal Distributions: Continuous Case
- 3.6 Independence of Random Variables
- 3.6.1 Independence of Discrete Random Variables
- 3.6.2 Independence of Continuous Random Variables
- 3.7 *Order Statistics
- 3.8 Problems
- 4 Operations on Random Variables
- 4.1 Mappings of Random Variables
- 4.2 Linear Transformations
- 4.3 Coin Tossing versus Uniform Distribution
- 4.3.1 Binary Fractions
- 4.3.2 Binary Fractions of Random Numbers
- 4.3.3 Random Numbers Generated by Coin Tossing
- 4.4 Simulation of Random Variables
- 4.5 Addition of Random Variables
- 4.5.1 Sums of Discrete Random Variables
- 4.5.2 Sums of Continuous Random Variables
- 4.6 Sums of Certain Random Variables
- 4.7 Products and Quotients of Random Variables
- 4.7.1 Student’s t-Distribution
- 4.7.2 F-Distribution
- 4.8 Problems
- 5 Expected Value, Variance, and Covariance
- 5.1 Expected Value
- 5.1.1 Expected Value of Discrete Random Variables
- 5.1.2 Expected Value of Certain Discrete Random Variables
- 5.1.3 Expected Value of Continuous Random Variables
- 5.1.4 Expected Value of Certain Continuous Random Variables
- 5.1.5 Properties of the Expected Value
- 5.2 Variance
- 5.2.1 Higher Moments of Random Variables
- 5.2.2 Variance of Random Variables
- 5.2.3 Variance of Certain Random Variables
- 5.3 Covariance and Correlation
- 5.3.1 Covariance
- 5.3.2 Correlation Coefficient
- 5.4 Problems
- 5.1 Expected Value
- 6 Normally Distributed Random Vectors
- 6.1 Representation and Density
- 6.2 Expected Value and Covariance Matrix
- 6.3 Problems
- 7 Limit Theorems
- 7.1 Laws of Large Numbers
- 7.1.1 Chebyshev’s Inequality
- 7.1.2 *Infinite Sequences of Independent Random Variables
- 7.1.3 * Borel–Cantelli Lemma
- 7.1.4 Weak Law of Large Numbers
- 7.1.5 Strong Law of Large Numbers
- 7.2 Central Limit Theorem
- 7.3 Problems
- 7.1 Laws of Large Numbers
- 8 Mathematical Statistics
- 8.1 Statistical Models
- 8.1.1 Nonparametric Statistical Models
- 8.1.2 Parametric Statistical Models
- 8.2 Statistical Hypothesis Testing
- 8.2.1 Hypotheses and Tests
- 8.2.2 Power Function and Significance Tests
- 8.3 Tests for Binomial Distributed Populations
- 8.4 Tests for Normally Distributed Populations
- 8.4.1 Fisher’s Theorem
- 8.4.2 Quantiles
- 8.4.3 Z-Tests or Gauss Tests
- 8.4.4 t-Tests
- 8.4.5 72-Tests for the Variance
- 8.4.6 Two-Sample Z-Tests
- 8.4.7 Two-Sample t-Tests
- 8.4.8 F-Tests
- 8.5 Point Estimators
- 8.5.1 Maximum Likelihood Estimation
- 8.5.2 Unbiased Estimators
- 8.5.3 Risk Function
- 8.6 Confidence Regions and Intervals
- 8.7 Problems
- 8.1 Statistical Models
- A Appendix
- A.1 Notations
- A.2 Elements of Set Theory
- A.3 Combinatorics
- A.3.1 Binomial Coefficients
- A.3.2 Drawing Balls out of an Urn
- A.3.3 Multinomial Coefficients
- A.4 Vectors and Matrices
- A.5 Some Analytic Tools
- Bibliography
- Index
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